The Spatial AutoRegressive model (SAR) is commonly used in studies involving spatial and network data to estimate the spatial or network peer influence and the effects of covariates on the response, taking into account the spatial or network dependence. While the model can be efficiently estimated with a Quasi maximum likelihood approach (QMLE), the detrimental effect of covariate measurement error on the QMLE and how to remedy it is currently unknown. If covariates are measured with error, then the QMLE may not have the $\sqrt{n}$ convergence and may even be inconsistent even when a node is influenced by only a limited number of other nodes or spatial units. We develop a measurement error-corrected ML estimator (ME-QMLE) for the parameters of the SAR model when covariates are measured with error. The ME-QMLE possesses statistical consistency and asymptotic normality properties. We consider two types of applications. The first is when the true covariate cannot be measured directly, and a proxy is observed instead. The second one involves including latent homophily factors estimated with error from the network for estimating peer influence. Our numerical results verify the bias correction property of the estimator and the accuracy of the standard error estimates in finite samples. We illustrate the method on a real dataset related to county-level death rates from the COVID-19 pandemic.

翻译：空间自回归模型（SAR）常用于涉及空间和网络数据的研究中，旨在估计空间或网络同伴影响以及协变量对响应变量的效应，同时考虑空间或网络依赖性。虽然该模型可通过拟极大似然估计法（QMLE）高效估计，但协变量测量误差对QMLE的有害影响及如何修正目前尚不明确。若协变量存在测量误差，QMLE可能失去√n收敛性，甚至在节点仅受有限其他节点或空间单元影响时产生不一致性。我们针对SAR模型在协变量存在测量误差时的参数估计问题，提出了一种测量误差校正的极大似然估计量（ME-QMLE）。该估计量具有统计一致性和渐近正态性。我们考虑了两种应用场景：其一为真实协变量无法直接测量、仅能观测代理变量的情况；其二为利用从网络中估计的潜在同质性因子（含误差）来估计同伴影响。数值结果验证了该估计量的偏差修正性质以及有限样本下标准误估计的准确性。我们通过COVID-19大流行期间县级死亡率的真实数据集演示了该方法的应用。